adamatzky a., jones, j. (2010) on electrical correlates of physarum polycephalum spatial activity....

ON ELECTRICAL CORRELATES OF

PHYSARUM POLYCEPHALUM

SPATIAL ACTIVITY:

CAN WE SEE PHYSARUM MACHINE IN THE DARK?

ANDREW ADAMATZKY AND JEFF JONES

Abstract. Plasmodium of Physarum polycephalum is a single cell visible by

unaided eye, which spans sources of nutrients with its protoplasmic network.

In a very simple experimental setup we recorded electric potential of the prop-agating plasmodium. We discovered a complex interplay of short range oscil-

latory behaviour combined with long range, low frequency oscillations which

serve to communicate information between different parts of the plasmod-ium. The plasmodium’s response to changing environmental conditions forms

basis patterns of electric activity, which are unique indicators of the following

events: plasmodium occupies a site, plasmodium functions normally, plasmod-ium becomes ‘agitated’ due to drying substrate, plasmodium departs a site,

and plasmodium forms sclerotium. Using a collective particle approximation

of Physarum polycephalum we found matching correlates of electrical potentialin computational simulations by measuring local population flux at the node

positions, generating trains of high and low frequency oscillatory behaviour.Motifs present in these measurements matched the response ‘grammar’ of the

plasmodium when encountering new nodes, simulated consumption of nutri-

ents, exposure to simulated hazardous illumination and sclerotium formation.The distributed computation of the particle collective was able to calculate

beneficial network structures and sclerotium position by shifting the active

growth zone of the simulated plasmodium. The results show future promisefor the non-invasive study of the complex dynamical behaviour within — and

health status of — living systems.

Keywords: Physarum polycephalum, plasmodium, electric potential, behaviour

1. Introduction

Plasmodium of Physarum polycephalum1 is a single cell with many diploid nuclei.The plasmodium feeds on microbial creatures and microscopic food particles. Whenplaced in an environment with distributed sources of nutrients the plasmodiumforms a network of protoplasmic tubes connecting the food sources. Nakagakiet al [24, 25, 26, 27] showed that the topology of the plasmodium’s protoplasmicnetwork optimizes the plasmodium’s harvesting on the scattered sources of nutrientsand makes more efficient flow and transport of intra-cellular components. Theseworks by Nakagaki et al. initiated the field of Physarum computing [9].

A plasmodium can be considered as large-scale collective of simple entities (micro-volumes, networks of bio-chemical oscillators) with distributed and massively-parallelsensing, computation and actuation. Sensing is parallel because the plasmodium

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can detect and determine the position of many sources of chemo-attractants, includ-ing nutrients, and also perform decentralized sensing of environmental conditions,including humidity, temperature and illumination. An actuation is parallel becausethe plasmodium can propagate in several directions in parallel, the plasmodiumcan occupy and colonize many food sources at the same. With regards to parallelcomputation, the plasmodium is a wave-based massively-parallel reaction-diffusionchemical computer [2, 3, 9]. A computation in the plasmodum is implementedby interacting bio-chemical and excitation waves [23], redistribution of electricalcharges on plasmodium’s membrane [1] and spatio-temporal dynamics of mechani-cal waves [23].

Experimental proofs of P. polycephalum computational abilities include approxi-mation of shortest path [26] and hierarchies of planar proximity graphs [5], computa-tion of plane tessellations [35], implementation of primitive memory [29], executionof basic logical computing schemes [37, 8], control of robot navigation [38], andnatural implementation of spatial logic and process algebra [30]. See an overviewof Physarum-based computers in [9].

In [4] we introduce a formalism of Physarum Machines by demonstrating thatplasmodium of P. polycephalum is a biological implementation of Kolmogorov-Uspenskii machine (KUM) [19, 20, 40]. A KUM is a storage modification machineoperating on a colored set of irregular graph nodes. The KUM is an predecessorto Knuth’s linking automata [21], Tarjan’s reference machine [36], and Schonhage’sstorage modification machines [32, 33]. Modern computers have an underlying ar-chitecture of random access machines, which are based on storage modificationmachines. Thus we can claim that Physarum Machines are biological implementa-tion of modern computers (indeed with drastically reduced functionality).

Localised propagating parts of plasmodium (pseudopodia, clusters of pseudopo-dia, wave-fragments) are analogs of active zones in the KUM, which in turn areequivalent to heads of Turing machines [19, 40]; see details in [4, 9].

In [9] we experimentally designed methods for physical control of the propaga-tion of active zones and thus demonstrated functionality and programmability ofPhysarum Machines. We demonstrated how to route active zones with attractingand repelling fields, merge active zones in one, multiply an active zone, translatean active zone from one data site to another, and direct active zone along specifiedvector [6, 7].

In all, publicized so far, Physarum-based computing devices data are representedby spatial distribution of attractants and repellents. Computation is implementedby foraging activity of plasmodium. And, results of computation are representedby configuration of plasmodium’s protoplasmic tubes, essentially by topology ofplasmodium’s body. That is results of computation are detected optically. Theoptical detection of computation results might be unfeasible when Physarum Ma-chines operate, either as standalone soft-body robots or being integrated in a hybridwetware-hardware systems, in concealed or non-transparent spaces. In present pa-per we explore one of the alternative methods for monitoring Physarum Machinesby using their electrical activity. We thus demonstrate a very simple techniquefor non-invasive monitoring of spatial development of plasmodium of Physarumpolycephalum.

On electrical correlates of Physarum polycephalum spatial activity 3

Assume we positioned an array of electrodes on bottom of a Petri dish, coveredelectrodes with agar blobs, inoculated the substrate with plasmodium and placed thedish into non-transparent box. We do not see the plasmodium. Can we reliablypredict the plasmodium’s spatial behavior in the dish by dynamics of its surfaceelectrical potential?

Early works on electrical activity of Physarum, dated back to Heilbrunn andDaugherty work on electric charge of protoplasmic colloids [12] (see also review [31]),were concerned with relation between electrical and peristaltic activities of proto-plasmic tubes of plasmodium. Thus, Kashimoto [18] found that normal oscillationof surface potential has amplitude about 5 mV and period 1.5-2 min [18]. Fingerleet al [11] provided evidences that average membrane potential -83.5 mV and thepotential shows no correlation with exposure to light. They also shown experi-mental indications confirming Meyer et al [22] proposition that calcium ion fluxthrough membrane triggers oscillators responsible for dynamic of contractile activ-ity. Exact characteristics of electric potential oscillations vary depending on stateof Physarum culture and experimental setups, see overview in [10]. Commonly ac-ceptable is that the potential oscillates with amplitude of 5 to 10 mV and period50-200 sec, associated with shuttle streaming of cytoplasm [22]. Similar character-istics were earlier hinted in the papers by Iwamura [13], and Kamiya and Abe [17].Said that a correlation between electrical and contractile oscillation of plasmodiumremain unclear and so far we can only accept a point that they both are governedby the same mechanism but may occur independently on each other, see overviewin [34].

Most early experiments on electrical activity of slime mould substantially reducedthe plasmodium’s freedom of action: in majority of the experiments a very short(few mm) single protoplasmic tube/vein fixed between two electrodes. Contrary, westudy electrical activity of a ‘free range’ plasmodium, just barely reduced to a seriesof agar blobs (covering electrodes) in a large experimental arena. This allows us tomonitor spatial activity of a Physarum Machine, without restricting its operationalcapabilities.

The paper is structured as follows. We overview experimental setup and a de-scription of a computational model in Sect. 2. Basic types of electrical activity,discovered in experiments, are presented in Sect. 3. Section 4 presents principalindicators of Physarum activity, which could be used in inferences of Physarumcalculi [30]. In Sect. 5 we present results using the particle model approximation ofPhysarum polycephalum of spatial and temporal activity which replicate the experi-mental findings, and also some potential limitations in the reconstruction of networkstructure from indirect historical measurements. Our results are summarized anddiscussed in Sect. 6.

2. Methods

Plasmodium of Physarum polycephalum was cultivated in plastic lunch boxes(with few holes punched in their lids for ventilation) on wet kitchen towels and fedwith oat flakes. Culture was periodically replanted to fresh substrate. Electrical ac-tivity of plasmodium was recorded with ADC-16 High Resolution Data Logger (PicoTechnology, UK) (Fig. 1a). In each experiment we arranged one ground/referenceelectrode and up to 8 recording electrodes on Petri dishes (9 cm in diameter) and


(a)

(b)

Figure 1. Experimental setup: (a) One reference and three mea-surement electrodes arranged on bottom of Petri dishes and con-nected to computer via ADC-16 data logger, (b) Close up photoof linearly-arranged electrodes, covered with agar blobs with oafflakes on top. Plasmodium is placed on rightmost agar blob, cov-ering reference/ground electrode.

covered them with blob of non-nutrient 2% agar gel. An oat flake was placed ontop of each agar blob to attract plasmodium and provide supply of nutrients. Atthe beginning of each experiment a piece of plasmodium was placed on agar blobcovering reference electrode (Fig. 1b).

To approximate the indirect inference of Physarum spatial behaviour we use theapproach introduced in [15] in which a population of simple, multi-agent based,mobile particles with chemotaxis-like sensory behaviour was used to construct andminimise spatially represented transport networks in a diffusive environment. Themethod corresponds to a particle approximation of LALI (Local Activation Long-range Inhibition) reaction-diffusion pattern formation processes [14] and a complexrange of patterning can be achieved by varying particle sensory parameters. Eachparticle in the collective represents a hypothetical unit of Physarum plasmodiumectoplasm and endoplasm which includes the effect of chemoattractant gradientson the plasmodium membrane (sensory behaviour) and the flow of protoplasmic


sol within the plasmodium (motor behaviour). The collective particle positions atany instance in time correspond to a static snapshot of dynamical network struc-ture whilst the collective movement of the particles in the network corresponds toprotoplasmic flow within the network.

Although the model is very simple in its assumptions and implementation it is ca-pable of reproducing some of the spontaneous network formation, network foraging,oscillatory behaviour, bi-directional shuttle streaming, and network adaptation seenin Physarum using only simple, local microscopic functionality to generate collec-tive macroscopic emergent behaviour. Details of the particle morphology, sensoryand motor behavioural algorithm can be found in [16] and a description of theoscillatory behaviour of the model can be found in [39]. In this paper we use anextension of the basic model to include plasmodium growth and adaptation (growthand shrinkage of the collective).

Growth and adaptation of the particle model population is currently imple-mented using a simple method based upon local measures of space availability(growth) and overcrowding (adaptation, or shrinkage, by population reduction).This is a gross simplification of the complex factors involved in growth and adapta-tion of the real organism (such as metabolic influences, nutrient conversion, wasteremoval, slime capsule coverage, bacterial contamination etc.). However the sim-plification renders the population growth and adaptation more computationallytractable and the specific parameters governing growth and shrinkage are at leastloosely based upon real environmental constraints of nutrient availability and diffu-sion. Growth and shrinkage states are iterated separately for each particle and theresults for each particle are indicated by tagging Boolean values to the particles.The method employed is specified as follows.

If there are 1 to 10 particles in a 9×9 neighbourhood of a particle, and the particlehas moved forwards successfully, the particle attempts to divide into two if there isan empty location in the immediate 3× 3 neighbourhood surrounding the particle.If there are 0 to 24 particles in a 5 × 5 neigbourhood of a particle the particlesurvives, otherwise it is annihilated. The frequency at which the growth/shrinkageof the population is executed determines a turnover rate for the particles. Lowerturnover rates (for example testing every 20 scheduler steps) resulted in longerpersistence of individual particles and stronger oscillatory readings from the nodessampling and was used for experiments with linear node arrays. For experimentsutilising more complex 2D spatial arrangements of nodes we used a higher turnoverrate (every ten scheduler steps) which results in more resilient network structure(i.e. the network paths are less likely to become fragmented into separate trees) atthe expense of oscillation signal strength.

To replicate the experimental spatial configuration the environment is repre-sented by a greyscale coded image isomorphic to the 2D lattice. Particular valuesrepresent uninhabitable boundaries, vacant areas (within which the population cangrow, move and adapt the collective morphology) and locations of nutrients. Aswith the experimental method the population was inoculated at a specific loca-tion in the environment and the spatial behaviour of the collective was indirectlyrecorded by sampling the activity at nutrient locations at every ten scheduler steps.One obvious difference between the experimental setup and the model is in the lim-itation of the model’s granularity. Because the chemical interactions within the


plasmodium membrane — which are ultimately responsible for the contractile be-haviour of the ectoplasm of the plasmodium — are not specifically modelled it isnot possible to record electrical potential. Instead we infer local activity of thecollective by measuring the population size within a fixed region window (11 × 11pixels in area.

Nutrient oat flakes were represented by projection of chemoattractant to the dif-fusion map at identical locations to the ‘electrodes’. The projection weight of thenode greyscale values (altering the strength of chemoattractant projection, higherweight values result in more chemoattractant projection to the environment) ap-plied to the nutrient locations was 0.1. Diffusion damping (limiting the distance ofchemoattractant gradient diffusion, lower values result in longer diffusion distances)was set to 0.001 and the diffusion kernel size was 7 × 7 pixels for all experiments.To prevent over-accumulation of chemoattractant in the diffusion map we depletedthe diffusion map by 0.01 units every scheduler step. We assumed that diffusion ofchemoattractant from nutrient locations was suppressed when the nutrients werecovered by particles. The suppression is implemented by checking each pixel atnutrient locations and reducing the projection value (i.e. concentration of chemoat-tractants), by multiplying it by 0.01 if any particles were within a 5 × 5 windowdistance of the nutrients.

Inoculation of the particle population was achieved by placing a small sample (be-tween 5 and 20 particles, depending on node size) on the start node. Particle sensoroffset was 5 pixels except where explicitly stated. Angle of rotation and sensor anglewere both set to 45 degrees in all experiments. Agent forward displacement was 1pixel per step and particles moving forwards successfully deposited 2.5 units into thediffusion map, resulting in a temporary storage of agent movement history in the dif-fusion map. Agents were subject to random influences on their choice of orientationangle and directional persistence of movement (p0.05 at every individual particlestep). The emergent transport networks are represented in the results by the con-figuration of the particle collective, shown as a spatial map of particle positions.The active zone can be seen as the most dense region of particles in the collective.Video recordings of the dynamical growth and adaptation and sclerotinisation of theparticle model can be found at http://uncomp.uwe.ac.uk/jeff/indirect.htm.

3. Patterns of plasmodium activity: experimental results

When growth of plasmodium was successful and no infestation with other mi-crobes occurred the blobs covering the electrodes became colonised by plasmodiumand connected in a chain by plasmodium’s protoplasmic tubes (Fig. 1b).

A typical situation is shown in Fig. 2. Plasmodium is inoculated on the referenceelectrode, and propagates onto recording electrode. Thus two electrodes becomeconnected by a single protoplasmic tube (Fig. 2a). Action potential (Fig. 2c, markedby triangle) indicates that plasmodium spreads from its original blob onto recordingelectrode blob. The potential rises from ground potential to 14 mV in 24 min,then drops down to roughly -5 mV in around 37 min. Polarization phase may beascribed to signaling on new source of nutrients to main body (still residing onreference electrode blob), while re-polarisation may be associated with propagationof plasmodium on recording electrode blob.

http://uncomp.uwe.ac.uk/jeff/indirect.htm


(a) (b)

(c)

Figure 2. Basic setup of experiments: (a) photo of two electrodesoccupied by plasmodia linked by a protoplasmic tube, (b) end ofexperiment, plasmodium dries up and forms sclerotium, (c) poten-tial on measurement electrode (channel 1) recorded during c. 43hours, Action potential is marked by triangle, typical oscillationsby ellipse (see Fig. 3), and sclerotinisation potential by rectangle.

After 8 hr transient period the system starts exhibiting regular oscillations ofpotential (Fig. 2d, encircled). The oscillations have amplitude 4-5 mV and fre-quency 30-40 min. Such type of oscillatory activity is observed in many electrodesetups. For example, the activity shown in (Fig. 4) comprised of oscillations withamplitude 3 mV and period 50-70 min. We believe the oscillations are associatedwith contractile waves running along the whole body of plasmodium and/or withsubstantial transfer of protoplasm along the electrode chain.


Figure 3. Typical oscillation of electric potential, marked by el-lipse in Fig. 2c.

Amplitude of oscillations is the same as in Kashimoto’s experiments [18] butperiod 15-20 times longer. A possible explanation could be that Kishimito measuredpotential differences between parts of plasmodium being 2-3 mm apart, while inour experiment distance can reach up to 2-3 cm. Thus it takes disturbances, whichrepresent peaks of electric potential, ten times longer to travel between electrodes.

In 23 hr from start of experiment agar blobs become dried, and the plasmodiumconverts itself into a sclerotium (Fig. 2d). The sclerotinisation is reflected in 50 mVpotential, which has the same amplitude as injury potential, described in [18],but much larger duration. Essentially the sclerotinisation potential drops onlywhen plasmodium completes its transformation to sclerotium (Fig. 2c, marked byrectangle).

In example illustrated in Fig. 5 plasmodium forms two protoplasmic tubes link-ing reference (original site of inoculation) and measurement electrode (colonised byplasmodium). It took plasmodium around 19 hr to recover after inoculation and topropagate from the reference electrode to measurement electrode (Fig. 5b). During10 days of recording various types of oscillatory activity have been observed, fromlow frequency waves with period 22-28 min and amplitude 3-4 mV to high-frequencywaves with period 1-2 min and amplitude 2-3 mV (Fig. 6). No high-frequency waveswere observed in the experiment with one protoplasmic tube (Fig. 2). This mayindicate that excitation waves travelling between two major plasmodium colonies(reference and measurement electrodes) cancel each other when collide. If two pro-toplasmic tubes are formed then a closed contour is formed and excitation wavescan travel in trains. The oscillatory patterns are super-imposed on very slow fluc-tuations of potential.


(a)

Figure 4. Very low frequency low amplitude oscillations. Thetrain of oscillations occurred in experiment shown in Fig. 9, channel2, during period 220000 sec to 252000 sec of experiment.

In some cases, e.g. marked by oval in Fig. 6, we observe a train of impulses leadsto slow polarization of the plasmodium on measurement electrode. In six impulses,each of amplitude c. 2 mV, the average potential increases by 10 mV. This followsby fast depolarisation and dropping of potential by 12 mV. This may indicatepropagation of a part of plasmodium from measurement to reference electrode.

Figure 7 helps us to build a detailed correspondence between spatial activityand electrical activity of plasmodium. When plasmodium occupies a blob of agaron top of an electrode a fast depolarisation is recorded on this electrode in mostcases (Fig. 7j, channels 2, 4,5). At this stage potential may drop down to -20 mV.The depolarisation is followed by fast polarisation, and the potential increases to0–5 mV. In some cases just very small (down to -3 mV) depolarisation occurswhen plasmodium colonises electrode. It follows by substantial polarisation, wherepotential may raise up to 10–15 mV (Fig. 7j, channels 4 and 7). Slow, i.e. spannedover a day or two, changes of potential difference on neighbouring electrodes canoccur in anti-phase (channels 5,6,7 in Fig. 7j gives more pronounced readings).These changes in potential reflect slow movement of large quantities of protoplasm(and possibly nutrients) between parts of plasmodium residing on neighbouringelectrodes.

With time, usually in 4-5 days, agar blobs became dry and not suitable forplasmodium. The plasmodium abandons the dry blobs. As clearly illustrated in


(a)

(b)

Figure 5. Example of two protoplasmic tubes connecting the elec-trodes: (a) photo of experimental dish at the stage when plasmod-ium colonised measuring electrode, on the left, (b) electrical activ-ity recorded for 10 days, (c) zoomed in part of the graph, showingoscillatory activity.


Figure 6. Oscillation of electric potential.

Fig. 7j, evacuation happens as an inverse of colonisation for channels 3 to 7. On thefifth day of experiment the plasmodium abandons electrodes 3 to 7 and concentrateson electrodes 1, 2 and 3 (Fig. 7h). At this stage we observe a succession of impulses(absolute amplitude 5-10 mV) on channels 1, 2 and 3 (Fig. 7j), which may reflecta decision-making process: the plasmodium choses on which one of three blobs toform a sclerotium. A blob on electrode 2 is chosen and the plasmodium completelymigrates onto this blob. Formation of sclerotium is reflected in steep rise in potentialto +22 mV (Fig. 7j, channel 2). At this moment the plasmodium shuts down allits physiological processes and electrical potential returns to 0 mV.

Occupation of an electrode by plasmodium is reflected in an impulse of, usually,negative charge (Fig. 8a). Sometimes we observe positive charge at the beginningof colonisation followed by decrease of voltage (Fig. 8b). Simultaneous colonisationof two or more electrodes cause synchronous changes in electrical potential on theseelectrodes. In experiment illustrated in Fig. 9 plasmodium spans reference electrodewith electrodes 1 and 2 at the same time (Fig. 9a). This is reflected as synchronousdepolarisation followed by polarisation (Fig. 9b, channels 1 and 2). Similar changesin electrical activity are observed when electrodes 3 and 4 are occupied (Fig. 9b,channels 3 and 4).

Drying of substrate causes high-amplitude oscillations of electrical activity, as-sociated with quick migration of plasmodium. Site where sclerotium is formed canbe detected by raise in electrical potential up to 20 mV, e.g. in examples shownin Fig. 8 sclerotium is formed on electrode 4 (channel 4, experiment Fig. 8a) andelectrode 2 (channel 2, experiment Fig. 8b).


(a) t = 0 sec (b) t = 70000 sec (c) t = 90000 sec

(d) t = 120000 sec (e) t = 190000 sec (f) t = 270000 sec

(g) t = 370000 sec (h) t = 450000 sec (i) t = 500000 sec

(j)

Figure 7. Snapshots of plasmodium propagating along chain ofelectrodes and electrical activity recorded. (a)–(i) photographsof the experimental dish, time lapsed from start of experiment isindicated in seconds, (j) electrical potential measured on sevenelectrodes.


(a)

(b)

Figure 8. Further examples of electrical activity of plasmodiumspanning chain of eight electrodes.


(a)

(b)

Figure 9. Electrical activity during simultaneous routing of plas-modium: (a) plasmodium spans reference electrode with electrodes1 to 4, (b) associated electrical activity.


Electrical activity Meaning Symbol

Arrival: Hello, I just came here αGood state: Thanks, I am OK ν

Agitation: I am troubled τ

Departure: Bye, I am leavingthis place

λ

Sclerotinisation: Good night, Iam going to sleep here

σ

Figure 10. Primitives of plasmodium’s communicative features,as expressed in its electric activity.

4. Principal indicators of Physarum electrical activity

The following events in plasmodium life can be non-invasively recorded by sub-substrate electrodes: plasmodium occupies a given site, plasmodium leaves the site,plasmodium functions normally, plasmodium is agitated, plasmodium is formingsclerotium.

Exemplar patterns of electrical activity and their common-sense interpretationsare shown in Fig. 10. Indeed there may be variations to duration and amplitudeof the pattern but their general structure remains characteristical. Thus, whenplasmodium just arrives onto an agar blob covering an electrode an action potentialis exhibited in an ideal situation, or at least a sharp drop in electric potentialfollowed by increase of the potential. When plasmodium leaves a recording electrodewe observe gradual and smooth decrease of electric potential, usually from positiveto ground.

In normal, undisturbed, physiological state the plasmodium exhibits classicaloscillations of electric potential, associated with the plasmodium’s peristaltic ac-tivity. In agitated state, e.g. when substrate starts to dry, plasmodium exhibitsirregular changes in electric potential. The irregularity is derived from combina-tion of several modes of oscillation and interaction of excitation waves generatedin distant parts of the plasmodium. Sclerotinisaton is associated with sharp rise inelectric potential, followed by sharp decline. Ascending part possibly correspondsto mobilisation of resources and preparation for hibernation, while descending partrepresents gradual shutting of all physiological systems.

Using notations, and ε for empty electrode/site, in Fig. 10 we can describe be-haviour of a plasmodium on a quasi-one-dimensional chain. Let us consider thefollowing examples (electrodes are numbered from the right, ε means empty node)in Fig. 11. This means that at time t = 1 plasmodium propagates on electrode 1.


4 3 2 1 tε ε ε ε 0ε ε ε α 1ε ε α ν 2ε α ν ν 3α ν ν ν 4ν ν ν ν 5τ τ τ τ 6λ τ τ τ 7ε λ τ λ 8ε ε σ ε 9ε ε ε ε 10

Figure 11. Examplar representation of plasmodium’s activity inexpressions of Fig. 10.

During time steps t = 2, 3 and 4 the plasmodium colonises electrodes 2 to 4. Attime step t = 5 the whole plasmodium is in normal physiological state. At timestep t = 6 substrates starts to dry and the plasmodium becomes agitated. Withsubstrate drying further the plasmodium decides to retract and vacate electrode 4at time step t = 7. Evacuation continues till time step t = 9, when the plasmodiumforms sclerotium at electrode 2. Electric potential of slerotium is zero, therefore nomore activity is observed in the chain of electrodes at time step t = 10.

How the above can be used in unconventional computation? We have alreadymentioned that plasmodium of P. polycephalum is an ideal substrate for bio-physicalimplementation of Kolmogorov-Uspenskii machine (KUM) [19, 20, 40]. In [4] weoutlined a theoretical paradigm of Physarum machines, which are realisation ofKolmogorov-Uspenskii machine in the plasmodium. Physarum machine is some-what equivalent to multi-head Turing machine on a graph. Active zone of Physarummachine is an analog of reading/writing head of Turing machine.

Behaviour of active zone of Physarum machines and/or heads of three-basedTuring machine can be uncovered by interpreting indicative expressions s of plas-modium in any given site i and its immediate neighbours i−1 and i+1 at time stept and previous time interval t−∆t. Thus a tuple necessary for interpretation willbe I = 〈st−∆t

i−1 st−∆ti st−∆t

i+1 sti〉. Any site i can be in one of six states {ε, α, ν, τ, λ, δ}(Fig. 10). Therefore there 1296 possible configuration of a tuple I, we will not bediscussing all of them here but provide below few exemplar interpretations:

• εεαα: active zone/head shifts from cell i+ 1 to cell i• νλνε: active zone/head shifts either to cell i− 1 or cell i+ 1• εεεα: active zone/head is externally positioned in cell i• τττσ: Physarum machine halts (computation completed).

5. Modelling plasmodium electrical activity

We initialised a small population at the leftmost node of a horizontal array ofeight nodes (Fig. 12a) and recorded the activity around the nodes by samplingthe population at each node every ten scheduler steps. The diffusion of nutrientsfrom the nodes stimulated the periphery of the collective which moved towards the


stimulus source. The movement of the collective generated free space within thecollective which caused the replication of particles and growth of the populationsize. As the collective spanned the nodes, the active zone surged forwards to engulfthe nutrients (Fig. 12b,c). Gradual consumption of the nutrients at the nodes wasimplemented by decrementing the nutrient values (starting at 255) by 0.1 per sched-uler step if any particles were present within a 5 × 5 window centred around eachnode. Consumption of the nutrients reduced the concentration gradients projectedat the nutrient sites until the cohesion of the collective was greater than the attrac-tion to the nodes, resulting in shrinkage of the collective (Fig. 12d–f) which mimicsthe sclerotium formation of Physarum in response to unfavourable environmentalconditions.

Plotting the traces of the node recordings (Fig. 12g) shows the activity aroundeach node. In contrast to the Physarum plasmodium there is a sharp increase inactivity whenever the collective encounters a node for the first time (‘Arrival’ state).The collective then settles into the classical oscillatory behaviour (‘Normal’) whilstnutrient levels remain high. Note that the amplitude of the oscillations is diminishedat the outermost node at the right (node 6) as this has less flux of particles throughit because it only has one node neighbour. As the concentration of the nutrientsfalls the activity around some — but not all — nodes declines slightly (for example(Fig. 12g, nodes 2,3,4 and 6) reproducing the ‘agitation’ pattern of the plasmodium.When the collective shrinks to form the pseudo-sclerotium there is a brief increasein activity at each node as the collective detaches and passes through the node,before a gradual fall in activity (‘departure’). Activity completely ceases at a nodewhen the collective is not within range. In (Fig. 12g) the sclerotium forms atnode 4, indicated by a characteristic rise in activity when all remaining nodes arequiescent (‘sclerotinisation’). This node remains minimally active, with a dormantpopulation size of between 70 and 80 particles.

Note that even if we could not observe the collective directly, it would be possibleto make some inferences about its spatial activity by considering the order of arrival(initial peak) and the order of detachment from each node. The final burst ofactivity in node 4 and remaining trace indicates that the sclerotium has settlednear this node.

In longer runs of the model where no consumption of nutrients occurred (as-suming a limitless supply of nutrients) and growth reached all nodes, we observed,after 4000 scheduler steps, regular very low frequency and high amplitude trains ofoscillation (period of approximately 3000 steps) superimposed on the backgroundoscillations (Fig. 13a and b). These peaks were caused by movement of the activezone resulting in large aggregation of particles at particular nodes (Fig. 13b andc) and were most commonly observed at the centre node when an odd number ofnodes was used, or alternating between the central position when an even number ofnodes was used. The phenomenon was strongest when the turnover rate of particleswas low (i.e. a low frequency of population growth/shrinkage, every 20 steps in thisexample). At higher particle turnover rates the effect was both less pronounced inamplitude and less predictable in its timing. Because high particle turnover ratestend to disrupt the propagation of oscillations within the collective we speculatethat the periodic peaks emerge due to a distributed computation arising from thelong term bulk transport of particles (and hence oscillations) within the network. Adistributed computation using such a mechanism would be an effective way for the


(a) t = 0 steps (b) t = 509 steps (c) t = 1252 steps

(d) t = 3870 steps (e) t = 4972 steps (f) t = 5232 steps

(g)

Figure 12. Snapshots of particle collective propagating alongchain of nutrient nodes and population activity recorded. (a) Ex-perimental design showing inoculation position (left), nodes andapproximate activity sampling region (circled) around each node,(b)and (c) Propagation of active zone of the collective as it growsto span the nutrient nodes, (d)–(f) Consumption of nutrients atthe nodes reduces stimuli causing the collective to shrink, approx-imating the sclerotium stage, (g) Activity at each node over time,lower traces correspond to leftwards position.

Physarum plasmodium to centralise its position efficiently between a large numberof nutrient sources.

Moving towards 2D growth of the population we were required to increase boththe sensor offset parameter from 5 pixels in distance to 9 pixels, and also increasethe turnover rate of the number of particles by increasing the frequency of pop-ulation growth/shrinkage to every 10 scheduler steps instead of every 20 steps.These changes are required to maintain network connectivity because the changetowards two dimensional movement of the active zone places greater stresses on theconnectivity of the network (since tension forces within the network structure are


(a)

(b)

(c) t = 9680 steps (d) t = 12380 steps

Figure 13. Emergence of strong low frequency periodic oscilla-tions within linear transport network. (a) Graph of activity oftransport network at the nodes of a 7 node linear chain showingstrong low frequency oscillations, (b) Enlarged portion of node 3activity illustrating large low frequency oscillation (15400-19400steps) superimposed over small amplitude high frequency oscilla-tions, (c) and (d) Visual appearance of network during strong lowfrequency peak shows aggregation of particles around nodes.

no longer limited to a single dimension). The simultaneous routing of the parti-cle collective generates a similar activity response as seen in the real plasmodium(Fig. 14). Parallel growth extends towards nodes simultaneously (Fig. 14b andc) and is reflected in the activity traces (Fig. 14g, dashed ovals indicating pairednodes). After 12000 steps both nodes 2 and 3 are exposed to simulated hazardouslight stimuli. Any particles within an irradiated region have their sensitivity tochemoattractant gradient reduced (by multiplying sensor values by 0.001). This


provokes the ‘agitated’ state (Fig. 14g, small dashed rectangle) and the connectionto the nodes is weakened as particles are repulsed by the hazard. When the hazardis removed there is once again an influx of particles to the nodes and the nodes arereconnected to the network. Consumption of all nutrient resources is initiated after18000 steps and sclerotinisation occurs when the nutrient gradients become weakerthan the mutual cohesion of the collective (Fig. 14g, large dashed rectangle). Assclerotinisation occurs outside any node region all activity traces are extinguished.The last node trace to become extinguished (node 3) is the final node through whichthe sclerotium forming collective passed through.

The experimental and modelling results previously shown suggest that informa-tion about the state of the plasmodium (based on the list of communication primi-tives) can be inferred non-invasively from the electrical potential (or bulk transportbased on population size) at specific sites in the environment. When coupled withcareful reference to the timing and history of plasmodium state changes (such astime of occupation, time of departure etc.), it is possible to infer an approximationof network structure and also distance between nodes (if we assume that diffusionof nutrients propagates at uniform speeds then detection of, and migration towards,nutrient nodes will reflect the distance between nodes). However the previous ex-amples given are limited in that they utilise one-dimensional networks or fairlyregular 2D arrays, as in the case of simultaneous contact between nodes in Fig. 9and Fig. 14. How well can we infer plasmodium behaviour — and possibly networkstructure — when a more complex 2D environment is used?

We patterned the model environment with a more complex arrangement of nodesand inoculated the particle model at one of the sites (Fig. 15a, node 0). Theparticle collective grew towards the nearest nutrient source (since the diffusion ofchemoattractant gradient is at uniform speed), engulfing the source, suppressingits diffusion of nutrients and thus exposing the active zone of the collective to thenutrients at the remaining nodes. The distributed computation afforded by thesuppression of gradients on contact and orientation towards new diffusion sourcesresulted in a complex translation of the position and orientation of the active growthzone of the collective as the transport network grew to span the remaining nodes(Fig. 15a—i). The final structure approximated the Steiner tree for this set of pointswhich represents the minimum amount of network material necessary to connectall nodes, thus the collective constructs an efficient transport network using thisdistributed means of computation. The resulting activity graph containing thecommunication primitives and timing information is shown in Fig. 15j. Can anyinformation about the network structure be inferred from the activity graph?

Tracking the connectivity order to indirectly create the network structure pro-vides an incorrect network structure when compared to the actual final networkformed (Fig. 17a and b respectively). This is because the active zone of the col-lective shifts position three times during the growth of the network. After growingto node 4 the active zone shifts to node 2 which then grows from this location andconnects to node 5. Similarly, after connecting node 6 the active zone shifts tonode 4 and extends to node 7 before shifting position to node 3 before growingtowards node 8. The result is a shorter network and one without crossover points,but how is this computed by the collective? The shift in active zone position isaccompanied by strong oscillations in nodes 2 and 3 (see Fig. 15j, nodes 2 and3). These nodes are located towards the centre position of the network and have




(g)

Figure 14. Simultaneous discovery of network nodes, hazard ex-posure and sclerotinisation. (a) Experimental design, inoculationpoint ‘s’ and node pairs 0–6, (b) and (c) Simultaneous discoveryof node pairs, (d) Repulsive light hazard is applied to rectangu-lar areas surrounding nodes 2 and 3, (e) and (f) Sclerotinisationoccurs when nutrients have been consumed, (g) Graph indicatingsimultaneous discovery of nodes (dashed ovals), agitation responsecaused by hazard exposure (small dashed rectangle), formation ofsclerotium outside of node regions.

degree of 3 connectivity at the end of the experiment (and hence greater flux ofparticles) when compared to the outer nodes in the tree which have a smaller degreeand less activity (Fig. 15j, remaining activity traces). The activity history of allnodes was visualised by calculating the mean activity of all nodes throughout anexperimental run. These mean values were scaled and plotted as 8-bit grey levelsin circles around the centre of each node, lighter values representing greater node




(g) t = 2309 steps (h) t = 5783 steps (i) t = 9052 steps

Figure 15. Dynamic transformation of active zone during growthand adaptation in more complex spatial arrangements. Growth ofcollective from inoculation point (circled), order of node arrival(numbered) and translation of active growth zone (arrowed).

activity. The regions were Gaussian blurred with a radius of 15 to replicate thediffusion from nodes and to ‘mix’ competing signals from nearby nodes (Fig. 17c)and the area of final sclerotium formation illustrated by the overlaid boxed region.The distributed computation thus automatically rewards nodes with greater flux ofparticles. One consequence of this is that when the nutrients are finally depletedthe sclerotium tends to form closer to nodes with greater flux - thus the collectivemanages to ‘hibernate’ near regions which have previously been rewarding in termsof nutrient availability and transport distance (Fig. 17d—f).


Figure 16. Graph indicating historical activity during growth inFig. 15, adaptation and sclerotinisation of particle model, nodes 2and 3 show strongest activity due to greater connectivity and fluxthrough these nodes.

(a) Contact order (b) Actual network (c) Activity history


Figure 17. Inference of network structure from active zone ar-rival time gives incorrect structure. (a) Network connectivity in-ferred from the active zone arrival time at nodes shows incorrectnetwork, (b) Actual final connectivity of network, (c) Illustra-tion of history of total activity at all nodes and final sclerotiumposition (boxed),(d)—(f) Sclerotium formation concentrates nearnodes with previously greater flux.


6. Discussion

We non-invasively monitored the state and activity of plasmodium stage of thetrue slime mould Physarum polycephalum by measuring its electrical activity usinga series of electrode probes placed at regular intervals in an unrestricted experimen-tal environment. We found that the electrical potential was affected by changes inenvironmental conditions and also by the growth, movement, adaptation and sclero-tinisation of the plasmodium itself. We observed relatively fast changes in potentialwhich were characteristic of the natural oscillation period of the plasmodium anddirect contact with stimuli, and slower changes which were consistent with thecontractile wave motion along the plasmodial network and the bulk transport ofprotoplasm within the plasmodium. A correspondence between directly observedspatial activity and electrical potential was observed and these patterns were foundto be consistent with particular events and/or behaviours. The small set of patternprimitives (‘arrival’, ‘normal’, ‘agitation’, ‘departure’ and ‘sclerotinisation’) facili-tate the non invasive monitoring of the organism and a limited record of overallactivity can be inferred and constructed using the historical electrode data. Theseprimitive expressions will be in, in our further papers, used for spatio-temporal rea-soning about Physarum machines, which are plasmodium-based implementationsof general-purpose storage-modification machines.

The simulation results from the particle approximation of Physarum plasmodiumsupport the hypothesis that a record of spatio-temporal activity can be constructedusing a limited number of indirect monitors — in this case the amount of ‘plasmod-ium’ in fixed windowed areas around each nutrient source to represent changes inpotential. We observed activity patterns which corresponded to the above patternprimitives in the experimental results. Low frequency and large amplitude changesin oscillatory behaviour in a linear sequence of nodes were observed which werefound to match visual changes in bulk transport of particles within the collective.The position of low frequency peaks of activity tended to be at the central positionin the node array, suggesting that a physical mechanism utilising competing wavepropagation was being utilised to determine this location. Long term bulk trans-port phenomena were also observed in changes to the active growth zone of thecollective in a complex 2D environment. We demonstrated that a perfect recordof the transport network could not be reconstructed via the historical record ofprimitive type and timing. Reconstructing the network from this record resultedin a less efficient pattern than was actually observed and the difference in structurewas found to be due to the internal and distributed computation which translatedthe position of the active growth zone. The result of active zone translation wasfound to maximise the flux and degree of connectivity at certain nodes at the centreof available nutrient sources. This also resulted in sclerotinisation occurring near apreviously strongly active nodes.

A plasmodium of Physarum polycephalum can be considered as a quasi-two-dimensional non-linear active medium encapsulated into a growing elastic mem-brane. Behaviour of the plasmodium is controlled or at least tune by travellingion-flux waves and contractile waves, and associated peristaltic transfer of cyto-plasm between distant part of the cell. Our techniques for non-invasive recordingof plasmodium’s activity and interpretation of results in terms of plasmodium’sspatial development is a valuable contribution to indirect diagnostics of spatially


extended non-linear media in physics, biology and chemistry. Controlling amor-phous robots will be yet another application domain for our methods. These willbe topics of further studies.

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Unconventional Computing Centre, University of the West of England,, BristolBS16 1QY, United Kingdom, {andrew.adamatzky, jeff.jones}@uwe.ac.uk

adamatzky a., jones, j. (2010) on electrical correlates of physarum polycephalum spatial activity....

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